Can I state the Mandelbrot Set/Equation like this?

In summary, the conversation discusses the equation f(z) = z^2 + c and its relation to the Mandelbrot set, which is the set of all complex numbers for which the iteration z_{n+1} = z_n^2 + c, z_0 = 0 converges. The question arises whether it is correct to write f(z) = z^2 + c or z(n+1) = z(n)^2 + c, and the answer depends on the context and understanding of the other person. The latter formula better captures the concept of an endless sequence of complex numbers.
  • #1
emergentecon
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f(z) = z2 + c
 
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  • #2
But it must be under iteration. Then definitely you can do that.
Here I suppose you are saying z and c are complex numbers.
 
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  • #3
Raghav Gupta said:
But it must be under iteration. Then definitely you can do that.
Here I suppose you are saying z and c are complex numbers.
Correct yes . . . under iteration, and z + c are complex numbers.
Thanks!
 
  • #4
Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"
 
  • #5
HallsofIvy said:
Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"

I can't say this helps me much.

Is it wrong to say that if you iterate the function f(z) = z^2 + c over the complex numbers, that you get the mandelbrot set?
 
  • #6
HallsofIvy said:
Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"

For instance, if I wanted to implement the recursion in Excel, I would in essence specify it as f(z) = z^2 + c
HallsofIvy said:
Just the simple equation "[itex]f(z)= z^2+ c[/itex]" is not a "definition" of anything! The Mandelbrot set is the set of all points, c, in the complex plane such that the recursion "[itex]z_{n+1}= z_n^2+ c[/itex], [itex]z_0= 0[/itex]"

The mandelbrot set is often 'summarised' as z = z^2 + c so why is it wrong to write f(z) = z^2 + c ?
 
  • #7
emergentecon said:
I can't say this helps me much.

Is it wrong to say that if you iterate the function f(z) = z^2 + c over the complex numbers, that you get the mandelbrot set?
? Just "iterating a function" will give you a lot of numbers and a lot of points where the iteration does not converge to any number. Again, the "Mandlebrot set" is the set of complex numbers for which the iteration does converge.

The mandelbrot set is often 'summarised' as z = z^2 + c so why is it wrong to write f(z) = z^2 + c ?
I don't know what you mean by "summarized" here. What do you do with that iteration to get the Mandlebrot set. (The "Julia sets" use that same iteration but in a different way.)
 
  • #8
Given f(z) = z2 + c then zn+1 = f(zn). z0=0, c ranges.

The mandelbrot set is over the domain, c.

If you want to produce a mandelbrot set, you need more than an equations. You need an algorithm.
 
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  • #9
emergentecon said:
The mandelbrot set is often 'summarised' as z = z^2 + c so why is it wrong to write f(z) = z^2 + c ?

It all depends on context. If you are talking to somebody about dynamical systems, then yes, you can write it like that. If you're not in the correct context, then no, you can't. It all depends on whether the other person will understand you or not.
 
  • #10
micromass said:
It all depends on context. If you are talking to somebody about dynamical systems, then yes, you can write it like that. If you're not in the correct context, then no, you can't. It all depends on whether the other person will understand you or not.
I mis-stated my question, meant to say his equation, as opposed to the set.
As I know, when he was asked about his work, he wrote down the equation as: z -> z^2 + c
So was wondering if, in this context, z = z^2 + c or f(z) = z^2 + c as opposed to z(n+1) = z(n)^2 + c
 
  • #11
emergentecon said:
I mis-stated my question, meant to say his equation, as opposed to the set.
As I know, when he was asked about his work, he wrote down the equation as: z -> z^2 + c
I would bet that it was z <- z2 + c, with the idea being that, starting with a specified complex number c and some complex number z, you square z, add c, and use that as your new z. Then repeat. And repeat. Ad infinitum.
emergentecon said:
So was wondering if, in this context, z = z^2 + c or f(z) = z^2 + c as opposed to z(n+1) = z(n)^2 + c
The latter formula better captures the idea of an endless sequence of complex numbers.
 
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  • #12
Thanks!
 

1. What is the Mandelbrot Set/Equation?

The Mandelbrot Set is a mathematical set of complex numbers that when iterated through the equation z = z^2 + c, produce a fractal pattern. The equation is named after mathematician Benoit Mandelbrot who discovered it in the 1970s.

2. Can I state the Mandelbrot Set/Equation in a simplified way?

Yes, the Mandelbrot Set can be stated as a set of complex numbers that, when plugged into the equation z = z^2 + c, do not diverge to infinity after a certain number of iterations.

3. Is there a specific way to plot the Mandelbrot Set/Equation?

Yes, the Mandelbrot Set is typically plotted on a complex plane where the real numbers make up the x-axis and the imaginary numbers make up the y-axis. The points that do not diverge to infinity after a certain number of iterations are colored black, while those that do are colored based on how quickly they diverge.

4. How is the Mandelbrot Set/Equation used in science?

The Mandelbrot Set has many applications in science, particularly in fields such as physics, biology, and computer science. It has been used to model natural phenomena such as coastlines and blood vessel networks, and to generate realistic computer graphics.

5. Can I change the values of the Mandelbrot Set/Equation?

Yes, the values of the Mandelbrot Set/Equation can be modified, such as by changing the constant c or the number of iterations. This can result in different fractal patterns and can also be used to study the behavior of the equation under different conditions.

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